# The intermediate value theory helps your table but not necessarily your beer

An Australian mathematician applied intermediate value theory to the problem of keeping a table from wobbling: Just rotate the table and you'll eventually find a spot.

A few things struck me about that article. First, that it explains that intermediate value theory "is the same principle underlying the fact that there will always be two points with exactly the same temperature somewhere on Earth." While that's true, it's way overkill. Intermediate value theory lets you find two points on any circle with the same temperature; you don't need a whole sphere. (Perhaps they got the scenario mixed up with the Borsuk-Ulam theorem, which shows the significantly stronger result that you can find two antipodal points on the Earth with the same temperature and pressure.)

The next thing that struck me is that the author twice used beer to illustrate the article. But then again, it was an Australian article, so maybe that's just normal for Australia.

Finally, the article concludes that the theorem may help you place your refrigerator. Well, sure, if you don't care which direction the door faces.

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1. richard says:

Hmmm … a three legged table would not have this problem.

While he might be able to solve the wobbling problem, this certainly does not address space efficiency since you can pack in more table if they are all neatly oriented rather than randomly oriented.

2. richard says:

Hmmm … a three legged table would not have this problem.

While he might be able to solve the wobbling problem, this certainly does not address space efficiency since you can pack in more table if they are all neatly oriented rather than randomly oriented.

3. Micah says:

Luckily fridges and washing machines come with adjustable feet.

4. BryanK says:

The first thing I thought was "that won’t help if the reason it’s wobbling is that one of its legs is too long".

If it’s wobbling because the floor is uneven, then rotating it will help.  But if it’s wobbling because one leg is two feet longer than the other three, then it won’t matter how much you rotate it.  You won’t be able to make it stop wobbling.  (The points at the ends of the table’s feet have to either be coplanar, or be close to coplanar.  Actually, it may not (mathematically) work at all if the feet aren’t exactly coplanar, but I’m not sure on that.)

5. dave says:

BryanK: "Actually, it may not (mathematically) work at all if the feet aren’t exactly coplanar, but I’m not sure on that."

A perfectly flat floor is an easy example where rotating the table will never work if the feet aren’t coplanar..

6. Niall says:

The placement of the refrigerator is another Australian thing. The placement of the door is unimportant, as long as the beer is cold.

7. mph says:

The use of beer as an example is not limited to Australians.  One of my astrophysics professors, who was Mexican-born, explained Rayleigh-Taylor instability with a Black-and-Tan poured in the wrong order.

8. Luke says:

Somebody has actually started to make money from wobbly tables:

http://news.bbc.co.uk/1/hi/england/surrey/6576005.stm

9. Ed says:

The actual paper is here:

http://arxiv.org/abs/math/0511490

And a video of the theory in action, featuring the world’s most uneven floor:

http://www.maths.monash.edu.au/~bpolster/table.mov

10. the Borsuk-Ulam theorem

Perhaps convoluted with the known result (I forget the attribution) that there are always at least two spots on Earth with no horizontal wind component.

11. Eric Lippert says:

How coincidental.  Just this past Sunday I put my circular, four-legged patio table out on my uneven lawn. As I do every year, I just kept rotating it until I found the spot where it didn’t wobble.

Applying the IVT to table legs sitting on top of a manifold of course requires a number of constraints upon the manifold that need to be called out if we’re going to treat this formally.  This technique wouldn’t work if my lawn were discontinuous, or not orientable, or if it had overhangs or holes.

12. Nicholas says:

there are always at least two spots on Earth

with no horizontal wind component.

That would be an application of the hairy ball theorem.

http://en.wikipedia.org/wiki/Hairy_ball_theorem

13. Dave says:

There’s a joke about a mathematician, a physicist, and an engineer here but I’m too lazy to write the whole thing. It ends with the engineer folding the paper with the mathematician’s proof and sticking it under one of the table legs.

14. Chris says:

A nit: it’s the Intermediate Value _Theorem_; calling it a “theory” makes it sound like it’s a whole subfield of mathematics (as defined here http://en.wikipedia.org/wiki/Theory#Mathematics).

[There are many intermediate value theorem*s*. The cited article adds to the body of results. That’s why I used the term “theory” – it’s the study of intermediate value theorems. -Raymond]
15. Will says:

"this certainly does not address space efficiency since you can pack in more table if they are all neatly oriented rather than randomly oriented"

This is why circular tables are popular in Australia. :)

16. Richard says:

It looks like this only works if the legs are equally, and arbitrarily, long. Plus, while the table won’t wobble, it won’t necessarily be *flat*. Nice idea, though :)

17. Steve D says:

I like the quote at the end:

”He admits that pure mathematics is usually associated with "ivory tower stuff".

"But every once in a while you can come up with something useful," he said.”

Like you said, as if this is useful in placing your fridge or washing machine…

Monash Uni is only a few blocks from my work.  Perhaps I should visit and see how his office furniture is arranged? And does he has a bar fridge in there? :-)

18. Igor says:

Does anyone thought of the fact that the room where you are placing the fridge has 4 sides and that if you always keep its back against the wall you will actually rotate it by 360° after trying all four walls?

On a sidenote, I would like to taste Australian beer, heard it is very good.

19. Kyralessa says:

Even if the table has one leg two feet longer than the others, it’ll still eventually sit level  if you rotate it because you’ll wear that leg even with the others.  You just have to rotate it for a very very long time.

<quote:BryanK>

"If it’s wobbling because the floor is uneven, then rotating it will help.  But if it’s wobbling because one leg is two feet longer than the other three, then it won’t matter how much you rotate it."

</quote:BryanK>

Actually, they are the same thing. If one leg is longer than other three, then, it will fit at some place which is deeper than the rest.

21. steveg says:

> The next thing that struck me is that the author twice used beer to illustrate the article. But then again, it was an Australian article, so maybe that’s just normal for Australia.

No, not really, the author’s an intellectual. A typical article would mention beer at least half-a-dozen times.

Free Beer & Wireless (from the Sydney Morning Herald, Sydney’s only *news*paper, the Telegraph is only good for wrapping broken beer bottles :-).

Oh crap I’m short. Beer, beer, beer, beer.

22. intermediate value says:

‘ ; drop beer ; select ‘

23. BryanK says:

Kyralessa:  Won’t two of the other three legs wear down at the same rate?

Oh, hold on, that might actually work.  Two of the other three legs will wear down at the same rate, but the fourth leg won’t, so eventually it’ll get to be long enough to make all four feet coplanar.  So yeah, that’s a solution.  I guess.  ;-)

Tanveer Badar: No, it won’t necessarily find a place which is "deeper than the rest".  Counterexample: the flat, planar floor.  It doesn’t matter how much you rotate the table (or move it, for that matter); you’ll never get the table to stop wobbling.  Whereas on an uneven (but continuous, orientable, etc.: see Eric Lippert’s comment) floor, a set of four coplanar feet will always be able to find a non-wobbling orientation by only rotating.  (Moving the table is not necessary here.)

There may be *some* floor topologies where a specific non-coplanar-feet set of four points can all be in contact with the floor at once.  But there are some floor topologies (e.g. the flat surface above) where that’s impossible.  And even the ones where it is possible will only work with a specific set of the four foot points; if you lengthen or shorten one leg, the surface is likely to stop working.

24. Igor says:

What if the floor wears too?

25. Nawak says:

The articles mentions "This explains three-legged stools are rarely wobbly"…

Hmm… Is this what happens when a journalist doesn’t believe an expert he is interviewing and adds "doubt" to what the expert says???

This explains why 2 and 2 rarely make 5.

(Except for very large values of 2 I guess)

26. Dave Harris says:

"Intermediate value theory lets you find two points on any circle with the same temperature; you don’t need a whole sphere."

Or presumably any closed path. And presumably you can find infinitely many such pairs. In fact, for any point you can presumably find another point at the same temperature, unless the first point happens to be the highest or the lowest temperature.

I am not arguing with anyone, but wanted to clarify some things.

<quote:BryanK>

"It doesn’t matter how much you rotate the table (or move it, for that matter); you’ll never get the table to stop wobbling."

</quote:BryanK>

For the flat surface, the intermediate value is the only value. If any leg (or variable) does not have that particular length (value), it can never be on that surface.

28. BryanK says:

> For the flat surface, the intermediate value is the only value. If any leg (or variable) does not have that particular length (value), it can never be on that surface.

Yes, that’s pretty much what I (and dave) said (although the legs don’t need to have the same length: just the feet need to be coplanar).  That’s why it’s not always possible to find an orientation where all four legs are on the surface, if the legs aren’t all the same length.  :-)

(*Sometimes* it is possible to find such an orientation, yes.  But it depends on both the floor surface’s shape and the specific lengths of the table legs; if you change either of those, then it may not be possible anymore.  You originally said "it will find some place that is deeper than the rest", but this is not always true: if there is no place that’s deeper than the rest, then the longer leg can’t find one.)

But maybe I’m just being pedantic…